Consider the expression \(3\mathrm{x}^2 - (2\mathrm{x}^2 - 5\mathrm{x} + 4) + (\mathrm{x} - 7)\).When the expression is simplified and written...

1. TRANSLATE the problem information

• Given expression: \(3\mathrm{x}^2 - (2\mathrm{x}^2 - 5\mathrm{x} + 4) + (\mathrm{x} - 7)\)
• Need to find: the coefficient b when written in form \(\mathrm{ax}^2 + \mathrm{bx} + \mathrm{c}\)

2. SIMPLIFY by distributing the negative sign

• The negative sign in front of the parentheses distributes to every term inside:

\(3\mathrm{x}^2 - (2\mathrm{x}^2 - 5\mathrm{x} + 4) + (\mathrm{x} - 7)\)

\(= 3\mathrm{x}^2 - 2\mathrm{x}^2 + 5\mathrm{x} - 4 + \mathrm{x} - 7\)

• Key insight: \(-(−5\mathrm{x})\) becomes \(+5\mathrm{x}\) (negative times negative equals positive)

3. SIMPLIFY by identifying and grouping like terms

• \(\mathrm{x}^2\) terms: \(3\mathrm{x}^2\) and \(-2\mathrm{x}^2\)
• \(\mathrm{x}\) terms: \(+5\mathrm{x}\) and \(+\mathrm{x}\)
• Constant terms: \(-4\) and \(-7\)

Grouped: \((3\mathrm{x}^2 - 2\mathrm{x}^2) + (5\mathrm{x} + \mathrm{x}) + (-4 - 7)\)

4. SIMPLIFY by combining each group

• \(\mathrm{x}^2\) terms: \(3\mathrm{x}^2 - 2\mathrm{x}^2 = 1\mathrm{x}^2 = \mathrm{x}^2\)
• \(\mathrm{x}\) terms: \(5\mathrm{x} + \mathrm{x} = 6\mathrm{x}\)
• Constants: \(-4 - 7 = -11\)

Final form: \(\mathrm{x}^2 + 6\mathrm{x} - 11\)

5. TRANSLATE to identify the coefficient

• In standard form \(\mathrm{ax}^2 + \mathrm{bx} + \mathrm{c}\): \(\mathrm{a} = 1\), \(\mathrm{b} = 6\), \(\mathrm{c} = -11\)
• Therefore \(\mathrm{b} = 6\)

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Poor SIMPLIFY execution: Sign error when distributing the negative sign, specifically with the \(-5\mathrm{x}\) term.

Students often write: \(3\mathrm{x}^2 - 2\mathrm{x}^2 - 5\mathrm{x} - 4 + \mathrm{x} - 7\) (keeping \(-5\mathrm{x}\) instead of \(+5\mathrm{x}\))

This leads to: \(\mathrm{x}^2 + (-5\mathrm{x} + \mathrm{x}) - 11 = \mathrm{x}^2 - 4\mathrm{x} - 11\)

They identify \(\mathrm{b} = -4\), leading them to guess since \(-4\) isn't an option, or potentially select Choice A (4) if they drop the negative sign.

Second Most Common Error:

Incomplete SIMPLIFY process: Forgetting to combine the x-terms from both parentheses.

Students correctly distribute signs but only combine the \(5\mathrm{x}\), missing the \(+\mathrm{x}\) from the last parentheses.

This gives: \(\mathrm{x}^2 + 5\mathrm{x} - 11\), so \(\mathrm{b} = 5\)

This leads them to select Choice B (5).

The Bottom Line:

This problem tests systematic algebraic manipulation. Success requires careful attention to signs during distribution and thorough combining of all like terms, not just the obvious ones.